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Chapter 4

Skill-biased technical change.Balanced growth theorems

This chapter is both an alternative and a supplement to the pages 60-64in Acemoglu, where the concepts of neutral technical change and balancedgrowth, including Uzawa’s theorem, are discussed.

Since “neutral” technical change should be seen in relation to “biased”technical change, Section 1 below introduces the concept of “biased” tech-nical change. Like regarding neutral technical change, also regarding biasedtechnical change there exist three different definitions, Hicks’, Harrod’s, andwhat the literature has dubbed “Solow’s”. Below we concentrate on Hick’sdefinition − with an application to the role of technical change for the evo-lution of the skill premium. So the focus is on the production factors skilledand unskilled labor rather than capital and labor. While regarding capitaland labor it is Harrod’s classifications that are most used in macroeconomics,regarding skilled and unskilled labor it is Hicks’.

The remaining sections discuss the concept of balanced growth and presentthree fundamental propositions about balanced growth. In view of the gen-erality of the propositions, they have a broad field of application. Our propo-sitions 1 and 2 are slight extensions of part 1 and 2, respectively, of whatAcemoglu calls Uzawa’s Theorem I (Acemoglu, 2009, p. 60). Our Proposi-tion 3 essentially corresponds to what Acemoglu calls Uzawa’s Theorem II(Acemoglu, 2009, p. 63).

57

58CHAPTER 4. SKILL-BIASED TECHNICAL CHANGE.

BALANCED GROWTH THEOREMS

4.1 The rising skill premium

4.1.1 Skill-biased technical change in the sense of Hicks:An example

Let aggregate output be produced through a differentiable three-factor pro-duction function ̃ :

= ̃ (1 2 )

where is capital input, 1 is input of unskilled labor (also called blue-collarlabor below), and 2 is input of skilled labor. Suppose technological changeis such that the production function can be rewritten

̃ (1 2 ) = ((1 2 )) (4.1)

where the “nested” function (1 2 ) represents input of a “human cap-ital” aggregate. Let be CRS-neoclassical w.r.t. and and let be CRS-neoclassical w.r.t. (1 2) Finally, let 0. So “technicalchange” amounts to “technical progress”.In equilibrium under perfect competition in the labor markets the relative

wage, often called the “skill premium”, will be

21=

21

=21

=2(1 2 )

1(1 2 )=

2(1 21 )

1(1 21 ) (4.2)

where we have used Euler’s theorem (saying that if is homogeneous ofdegree one in its first two arguments, then the partial derivatives of arehomogeneous of degree zero w.r.t. these arguments).Time is continuous (nevertheless the time argument of a variable, is in

this section written as a subscript ). Hicks’ definitions are now: If for all21 0

³2(121)1(121)

´

21 constant T 0 then technical change is⎧⎨⎩ skill-biased in the sense of Hicks,skill-neutral in the sense of Hicks.blue collar-biased in the sense of Hicks,(4.3)respectively.In the US the skill premium (measured by the wage ratio for college

grads vis-a-vis high school grads) has had an upward trend since 1950 (see

c° Groth, Lecture notes in Economic Growth, (mimeo) 2015.

4.1. The rising skill premium 59

for instance Jones and Romer, 2010).1 If in the same period the relativesupply of skilled labor had been roughly constant, by (17.3) in combinationwith (17.2), a possible explanation could be that technological change hasbeen skill-biased in the sense of Hicks. In reality, in the same period also therelative supply of skilled labor has been rising (in fact even faster than theskill premium). Since in spite of this the skill premium has risen, it suggeststhat the extend of “skill-biasedness” has been even stronger.Wemay alternatively put it this way. As the function is CRS-neoclassical

w.r.t. 1 and 2 we have 22 0 and 12 0 cf. Chapter 2. Hence, by(17.2), a rising 21 without technical change would imply a declining skillpremium. That the opposite has happened must, within our simple model,be due to (a) there has been technical change, and (b) technical changehas favoured skilled labor (which means that technical change has been skill-biased in the sense of Hicks).An additional aspect of the story is that skill-biasedness helps explain

the observed increase in the relative supply of skilled labor. If for a constantrelative supply of skilled labor, the skill premium is increasing, this increasestrengthens the incentive to go to college. Thereby the relative supply ofskilled labor (reflecting the fraction of skilled labor in the labor force) tendsto increase.

4.1.2 Capital-skill complementarity

An additional potential source of a rising skill premium is capital-skill com-plementarity. Let the aggregate production function be

= ̃ (1 2 ) = (11 22) = (+11)(22)

1− 0 1

where 1 and 2 are technical coefficients that may be rising over time.In this production function capital and unskilled labor are perfectly substi-tutable (the partial elasticity of factor substitution between them is +∞) Onthe other hand there is direct complementarity between capital and skilledlabor, i.e., 2(2) 0Under perfect competition the skill premium is

21

=22

=( +11)

(1− )(22)−2( +11)−11(22)1−

(4.4)

=1−

µ +1122

¶21

1On the other hand, over the years 1915 - 1950 the skill premium had a downwardtrend (Jones and Romer, 2010).

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60CHAPTER 4. SKILL-BIASED TECHNICAL CHANGE.

BALANCED GROWTH THEOREMS

Here, if technical change is absent (1 and 2 constant), a rising capitalstock will, for fixed 1 and 2 raise the skill premium.A more realistic scenario is, however, a situation with an approximately

constant real interest rate, cf. Kaldor’s stylized facts. We have, again byperfect competition,

= ( +11)

−1(22)1− = µ +1122

¶−1= + (4.5)

where is the real interest rate at time and is the (constant) capitaldepreciation rate. For = a constant, (17.5) gives

+1122

=

µ +

¶− 11−≡ (4.6)

a constant. In this case, (17.4) shows that capital-skill complementarity isnot sufficient for a rising skill premium. A rising skill premium requires thattechnical change brings about a rising 21. So again an observed risingskill premium, along with a more or less constant real interest rate, suggeststhat technical change is skill-biased.We may rewrite (4.6) as

22= − 11

22

where the conjecture is that 11(22) → 0 for → ∞ The analysissuggests the following story. Skill-biased technical progress generates risingproductivity as well as a rising skill premium. The latter induces more andmore people to go to college. The rising level of education in the labor forceraises productivity further. This is a basis for further capital accumulation,continuing to replace unskilled labor, and so on.In particular since the early 1980s the skill premium has been sharply

increasing in the US (see Acemoglu, p. 498). This is also the period whereICT technologies took off.

4.2 Balanced growth and constancy of key ra-tios

The focus now shifts to homogeneous labor vis-a-vis capital.We shall state general definitions of the concepts of “steady state” and

“balanced growth”, concepts that are related but not identical. With respect

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4.2. Balanced growth and constancy of key ratios 61

to “balanced growth” this implies a minor deviation from the way Acemoglubriefly defines it informally on his page 57. The main purpose of the presentchapter is to lay bare the connections between these two concepts as wellas their relation to the hypothesis of Harrod-neutral technical progress andKaldor’s stylized facts.

4.2.1 The concepts of steady state and balanced growth

A basic equation in many one-sector growth models for a closed economy incontinuous time is

̇ = − = − − ≡ − (4.7)where is aggregate capital, aggregate gross investment, aggregateoutput, aggregate consumption, aggregate gross saving (≡ −), and ≥ 0 is a constant physical capital depreciation rate.Usually, in the theoretical literature on dynamic models, a steady state is

defined in the following way:

Definition 3 A steady state of a dynamic model is a stationary solution tothe fundamental differential equation(s) of the model.

Or briefly: a steady state is a stationary point of a dynamic process.Let us take the Solow growth model as an example. Here gross saving

equals where is a constant, 0 1 Aggregate output is given by aneoclassical production function, with CRS and Harrod-neutral technicalprogress: = () = (̃ 1) ≡ (̃) where is the laborforce, is the level of technology, and ̃ ≡ () is the (effective) capitalintensity. Moreover, 0 0 and 00 0 Solow assumes () = (0) and() = (0), where ≥ 0 and ≥ 0 are the constant growth rates of thelabor force and technology, respectively. By log-differentiating ̃ w.r.t. 2

we end up with the fundamental differential equation (“law of motion”) ofthe Solow model: ·

̃ = (̃)− ( + + )̃ (4.8)Thus, in the Solow model, a (non-trivial) steady state is a ̃∗ 0 such that,

if ̃ = ̃∗ then·̃ = 0 In passing we note that, by (4.8), such a ̃∗ must

satisfy the equation (̃∗)̃∗ = ( + + ) and in view of 00 0 it isunique if it exists.The most common definition in the literature of balanced growth for an

aggregate economy is the following:2Or by directly using the fraction rule, see Appendix A to Chapter 3.

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62CHAPTER 4. SKILL-BIASED TECHNICAL CHANGE.

BALANCED GROWTH THEOREMS

Definition 4 A balanced growth path is a path ()∞=0 along which thequantities and are positive and grow at constant rates (not necessarilypositive and not necessarily the same).

Acemoglu, however, defines (Acemoglu, 2009, p. 57) balanced growthin the following way: “balanced growth refers to an allocation where outputgrows at a constant rate and capital-output ratio, the interest rate, and factorshares remain constant”. My problem with this definition is that it mixesgrowth of aggregate quantities with income distribution aspects (interest rateand factor income shares). And it is not made clear what is meant by theoutput-capital ratio if the relative price of capital goods is changing overtime. So I stick to the definition above which is quite standard and is knownto function well in many different contexts.Note that in the Solow model (as well as in many other models) we have

that if the economy is in a steady state, ̃ = ̃∗ then the economy featuresbalanced growth. Indeed, a steady state of the Solow model implies bydefinition that ̃ ≡ () is constant. Hence must grow at the sameconstant rate as namely + In addition, = (̃∗) in a steadystate, showing that also must grow at the constant rate + And somust then = (1 − ) So in a steady state of the Solow model the pathfollowed by ()∞=0 is a balanced growth path.As we shall see in the next section, in the Solow model (and many other

models) the reverse also holds: if the economy features balanced growth,then it is in a steady state. But this equivalence between steady state andbalanced growth does not hold in all models.

4.2.2 A general result about balanced growth

An interesting fact is that, given the dynamic resource constraint (4.7), wehave always that if there is balanced growth with positive gross saving, thenthe ratios and are constant (by “always” is meant: indepen-dently of how saving is determined and of how the labor force and technologyevolve). And also the other way round: as long as gross saving is positive,constancy of the and ratios is enough to ensure balanced growth.So balanced growth and constancy of key ratios are essentially equivalent.This is a very practical general observation. And since Acemoglu does not

state any balanced growth theorem at this general level, we shall do it here,together with a proof. Letting denote the growth rate of the (positivelyvalued) variable i.e., ≡ ̇ we claim:

Proposition 1 (the balanced growth equivalence theorem). Let ()∞=0

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4.2. Balanced growth and constancy of key ratios 63

be a path along which , and ≡ − are positive for all ≥ 0Then, given the accumulation equation (4.7), the following holds:

(i) if there is balanced growth, then = = and the ratios and are constant;

(ii) if and are constant, then and grow at the sameconstant rate, i.e., not only is there balanced growth, but the growthrates of and are the same.

Proof Consider a path ()∞=0 along which , and ≡ − are positive for all ≥ 0 (i) Assume there is balanced growth. Then, bydefinition, and are constant. Hence, by (4.7), we have that = + is constant, implying

= (*)

Further, since = +

=̇

=

̇

+

̇

=

+

=

+

(by (*))

=

+

−

=

( − ) + (**)

Now, let us provisionally assume that 6= Then (**) gives

=

− − (***)

which is a constant since and are constant. Constancy of requires that = hence, by (***), = 1 i.e., = In viewof = + , however, this outcome contradicts the given condition that 0 Hence, our provisional assumption and its implication, (***), arefalsified. Instead we have = . By (**), this implies = = butnow without the condition = 1 being implied. It follows that and are constant.(ii) Suppose and are constant. Then = = , so that

is a constant. We now show that this implies that is constant.Indeed, from (4.7), = 1− so that also is constant. It followsthat = = so that is constant. By (4.7),

=

̇ +

= +

so that is constant. This, together with constancy of and implies that also and are constant. ¤

c° Groth, Lecture notes in Economic Growth, (mimeo) 2015.

64CHAPTER 4. SKILL-BIASED TECHNICAL CHANGE.

BALANCED GROWTH THEOREMS

Remark. It is part (i) of the proposition which requires the assumption 0for all ≥ 0 If = 0 we would have = − and ≡ − = hence = for all ≥ 0 Then there would be balanced growth if the commonvalue of and had a constant growth rate. This growth rate, however,could easily differ from that of Suppose = 1− = and = ( and constants). Then we would have = = −+(1−) whichcould easily be strictly positive and thereby different from = − ≤ 0 sothat (i) no longer holds. ¤

The nice feature is that this proposition holds for any model for whichthe simple dynamic resource constraint (4.7) is valid. No assumptions aboutfor example CRS and other technology aspects or about market form areinvolved. Note also that Proposition 1 suggests a link from balanced growthto steady state. And such a link is present in for instance the Solow model.Indeed, by (i) of Proposition 1, balanced growth implies constancy of which in the Solow model implies that (̃)̃ is constant. In turn, the latteris only possible if ̃ is constant, that is, if the economy is in steady state.There exist cases, however, where this equivalence does not hold (some

open economy models and some models with embodied technological change,see Groth et al., 2010). Therefore, it is recommendable always to maintaina distinction between the terms steady state and balanced growth.

4.3 The crucial role of Harrod-neutrality

Proposition 1 suggests that if one accepts Kaldor’s stylized facts (see Chapter1) as a characterization of the past century’s growth experience, and if onewants a model consistent with them, one should construct the model suchthat it can generate balanced growth. For a model to be capable of generatingbalanced growth, however, technological progress must be of the Harrod-neutral type (i.e., be labor-augmenting), at least in a neighborhood of thebalanced growth path. For a fairly general context (but of course not asgeneral as that of Proposition 1), this was shown already by Uzawa (1961).We now present a modernized version of Uzawa’s contribution.Let the aggregate production function be

() = ̃ (() () ) 0 (4.9)

where is a constant that depends on measurement units. The only tech-nology assumption needed is that ̃ has CRS w.r.t. the first two arguments(̃ need not be neoclassical for example). As a representation of technical

c° Groth, Lecture notes in Economic Growth, (mimeo) 2015.

4.3. The crucial role of Harrod-neutrality 65

progress, we assume ̃ 0 for all ≥ 0 (i.e., as time proceeds, un-changed inputs result in more and more output). We also assume that thelabor force evolves according to

() = (0) (4.10)

where is a constant. Further, non-consumed output is invested and so (4.7)is the dynamic resource constraint of the economy.

Proposition 2 (Uzawa’s balanced growth theorem) Let = ( () () ())∞=0,where 0 () () for all ≥ 0 be a path satisfying the capital accumu-lation equation (4.7), given the CRS-production function (4.9) and the laborforce path in (4.10). Then:

(i) a necessary condition for this path to be a balanced growth path is thatalong the path it holds that

() = ̃ (() () ) = ̃ (() ()() 0) (4.11)

where () = with ≡ − ;(ii) for any 0 such that there is a + + with the property

that the production function ̃ in (4.9) allows an output-capital ratioequal to at = 0 (i.e., ̃ (1 ̃−1 0) = for some real number ̃ 0),a sufficient condition for the path P to be a balanced growth path withoutput-capital ratio , is that the technology can be written as in (4.11)with () = .

Proof (i)3 Suppose the path ( ()() ())∞=0 is a balanced growth path.By definition, and are then constant, so that () = (0) and () = (0) We then have

()− = (0) = ̃ ((0) (0) 0) = ̃ (()− ()− 0) (*)

where we have used (4.9) with = 0 In view of the precondition that ()≡ ()−() 0 we know from (i) of Proposition 1, that is constantso that = . By CRS, (*) then implies

() = ̃ (() − () − 0) = ̃ (() ( −)() 0)

We see that (4.11) holds for () = with ≡ − 3This part draws upon Schlicht (2006), who generalized a proof in Wan (1971, p. 59)

for the special case of a constant saving rate.

c° Groth, Lecture notes in Economic Growth, (mimeo) 2015.

66CHAPTER 4. SKILL-BIASED TECHNICAL CHANGE.

BALANCED GROWTH THEOREMS

(ii) Suppose (4.11) holds with () = Let 0 be given such thatthere is a + + with the property that

̃ (1 ̃−1 0) = (**)

for some constant ̃ 0 Our strategy is to prove the claim in (ii) by con-struction of a path = ( ()() ())∞=0 which satisfies it. We let be such that the saving-income ratio is a constant ≡ ( + + ), i.e., ()−() ≡ () = () for all ≥ 0 Inserting this, together with () =(̃())()(), where (̃()) ≡ ̃ (̃() 1 0) and ̃() ≡ ()(()())into (4.7), we get the Solow equation (4.8). Hence ̃() is constant if andonly if ̃() satisfies the equation (̃())̃() = ( + + ) ≡ By (**)and the definition of the required value of ̃() is ̃ which is thus thesteady-state for the constructed Solow equation. Letting (0) satisfy (0)= ̃(0) where = (0) we thus have ̃(0) = (0)((0)(0)) = ̃ Sothat the initial value of ̃() equals the steady state value. It now followsthat ̃() = ̃ for all ≥ 0 and so ()() = (̃())̃() = (̃)̃ = for all ≥ 0 In addition, () = (1− ) () so that () () is constantalong the path By (ii) of Proposition 1 now follows that the path is abalanced growth path, as was to be proved. ¤

The form (4.11) indicates that along a balanced growth path, technicalprogress must be purely “labor augmenting”, that is, Harrod-neutral. It is inthis case convenient to define a new CRS function, by (() ()())≡ ̃ (() ()() 0) Then (i) of the proposition implies that at least alongthe balanced growth path, we can rewrite the production function this way:

() = ̃ (() (0)() ) = (() ()()) (4.12)

where (0) = () = (0) with ≡ − It is important to recognize that the occurrence of Harrod-neutrality says

nothing about what the source of technological progress is. Harrod-neutralityshould not be interpreted as indicating that the technological progress em-anates specifically from the labor input. Harrod-neutrality only means thattechnical innovations predominantly are such that not only do labor and cap-ital in combination become more productive, but this happens to manifestitself at the aggregate level in the form (4.12).4

What is the intuition behind the Uzawa result that for balanced growth tobe possible, technical progress must have the purely labor-augmenting form?

4For a CRS Cobb-Douglas production function with technological progress, Harrod-neutrality is present whenever the output elasticity w.r.t capital (often denoted ) isconstant over time.

c° Groth, Lecture notes in Economic Growth, (mimeo) 2015.

4.3. The crucial role of Harrod-neutrality 67

First, notice that there is an asymmetry between capital and labor. Capitalis an accumulated amount of non-consumed output. In contrast, in simplemacro models labor is a non-produced production factor which (at least inthe context of (4.10)) grows in an exogenous way. Second, because of CRS,the original formulation, (4.9), of the production function implies that

1 = ̃ (()

()()

() ) (4.13)

Now, since capital is accumulated non-consumed output, it tends to inheritthe trend in output such that () () must be constant along a balancedgrowth path (this is what Proposition 1 is about). Labor does not inherit thetrend in output; indeed, the ratio () () is free to adjust as time proceeds.When there is technical progress (̃ 0) along a balanced growth path,this progress must manifest itself in the form of a changing () () in (13.5)as proceeds, precisely because () () must be constant along the path.In the “normal” case where ̃ 0 the needed change in () () is afall (i.e., a rise in ()()) This is what (13.5) shows. Indeed, the fall in() () must exactly offset the effect on ̃ of the rising when there is afixed capital-output ratio.5 It follows that along the balanced growth path, ()() is an increasing implicit function of If we denote this function() we end up with (4.12) with specified properties ( and ).The generality of Uzawa’s theorem is noteworthy. The theorem assumes

CRS, but does not presuppose that the technology is neoclassical, not tospeak of satisfying the Inada conditions.6 And the theorem holds for exoge-nous as well as endogenous technological progress. It is also worth mentioningthat the proof of the sufficiency part of the theorem is constructive. It pro-vides a method to construct a hypothetical balanced growth path (BGP fromnow).7

A simple implication of the Uzawa theorem is the following. Interpretingthe () in (4.11) as the “level of technology”, we have:

COROLLARY Along a BGP with positive gross saving and the technologylevel, () growing at the rate output grows at the rate + while laborproductivity, ≡ and consumption per unit of labor, ≡ growat the rate

5This way of presenting the intuition behind the Uzawa result draws upon Jones andScrimgeour (2008).

6Many accounts of the Uzawa theorem, including Jones and Scrimgeour (2008), presumea neoclassical production function, but the theorem is much more general.

7Part (ii) of Proposition 2 is left out in Acemoglu’s book.

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68CHAPTER 4. SKILL-BIASED TECHNICAL CHANGE.

BALANCED GROWTH THEOREMS

Proof That = + follows from (i) of Proposition 2. As to the growthrate of labor productivity we have

= (0)

(0)= (0)( −) = (0)

Finally, by Proposition 1, along a BGP with 0 must grow at the samerate as ¤We shall now consider the implication of Harrod-neutrality for the income

shares of capital and labor when the technology is neoclassical and marketsare perfectly competitive.

4.4 Harrod-neutrality and the functional in-come distribution

There is one facet of Kaldor’s stylized facts we have so far not related toHarrod-neutral technical progress, namely the long-run “approximate” con-stancy of both the income share of labor, and the rate of return tocapital. At least with neoclassical technology, profit maximizing firms, andperfect competition in the output and factor markets, these properties areinherent in the combination of constant returns to scale, balanced growth,and the assumption that the relative price of capital goods (relative to con-sumption goods) is constant over time. The latter condition holds in modelswhere the capital good is nothing but non-consumed output, cf. (4.7).8

To see this, we start out from a neoclassical CRS production functionwith Harrod-neutral technological progress,

() = (() ()()) (4.14)

With () denoting the real wage at time in equilibrium under perfectcompetition the labor income share will be

()()

()=

()()

()

()=

2(() ()())()()

() (4.15)

In this simple model, without natural resources, (gross) capital income equalsnon-labor income, () − ()() Hence, if () denotes the (net) rate ofreturn to capital at time , then

() = ()− ()()− ()

() (4.16)

8The reader may think of the “corn economy” example in Acemoglu, p. 28.

c° Groth, Lecture notes in Economic Growth, (mimeo) 2015.

4.4. Harrod-neutrality and the functional income distribution 69

Denoting the (gross) capital income share by () we can write this ()(in equilibrium) in three ways:

() ≡ ()− ()() ()

=(() + )()

()

() = (() ()())− 2(() ()())()()

()=

1(() ()())()

()

() =

()()

()

() (4.17)

where the first row comes from (4.16), the second from (4.14) and (4.15), thethird from the second together with Euler’s theorem.9 Comparing the firstand the last row, we see that in equilibrium

()

()= () +

In this condition we recognize one of the first-order conditions in the rep-resentative firm’s profit maximization problem under perfect competition,since () + can be seen as the firm’s required gross rate of return.10

In the absence of uncertainty, the equilibrium real interest rate in thebond market must equal the rate of return on capital, () And () + canthen be seen as the firm’s cost of disposal over capital per unit of capital pertime unit, consisting of interest cost plus capital depreciation.

Proposition 3 (factor income shares and rate of return under balancedgrowth) Let the path (() () ())∞=0 be a BGP in a competitive economywith the production function (4.14) and with positive saving. Then, along theBGP, the () in (4.17) is a constant, ∈ (0 1). The labor income sharewill be 1− and the (net) rate of return on capital will be = − where is the constant output-capital ratio along the BGP.

Proof By CRS we have () = (() ()()) = ()() (̃() 1)≡ ()()(̃()) In view of part (i) of Proposition 2, by balanced growth, ()() is some constant, . Since ()() = (̃())̃() and 00 0this implies ̃() constant, say equal to ̃∗ But ()() = 0(̃()) which

9From Euler’s theorem, 1 + 2 = () when is homogeneous of degreeone10With natural resources, say land, entering the set of production factors, the formula,

(4.16), for the rate of return to capital should be modified by subtracting land rents fromthe numerator.

c° Groth, Lecture notes in Economic Growth, (mimeo) 2015.

70CHAPTER 4. SKILL-BIASED TECHNICAL CHANGE.

BALANCED GROWTH THEOREMS

then equals the constant 0(̃∗) along the BGP. It then follows from (4.17)that () = 0(̃∗) ≡ Moreover, 0 1 where 0 follows from 0 0 and 1 from the fact that = = (̃∗)̃∗ 0(̃∗) in viewof 00 0 and (0) ≥ 0 Then, by the first equality in (4.17), ()() ()= 1− () = 1− . Finally, by (4.16), the (net) rate of return on capital is = (1− ()() ()) ()()− = − ¤

This proposition is of interest by displaying a link from balanced growthto constancy of factor income shares and the rate of return, that is, someof the “stylized facts” claimed by Kaldor. Note, however, that although theproposition implies constancy of the income shares and the rate of return,it does not determine them, except in terms of and But both and,generally, are endogenous and depend on ̃∗11 which will generally beunknown as long as we have not specified a theory of saving. This takes usto theories of aggregate saving, for example the simple Ramsey model, cf.Chapter 8 in Acemoglu’s book.

4.5 What if technological change is embod-ied?

In our presentation of technological progress above we have implicitly as-sumed that all technological change is disembodied. And the way the propo-sitions 1, 2, and 3, are formulated assume this.As noted in Chapter 2, disembodied technological change occurs when new

technical knowledge advances the combined productivity of capital and laborindependently of whether the workers operate old or new machines. Consideragain the aggregate dynamic resource constraint (4.7) and the productionfunction (4.9):

̇() = ()− () (4.18) () = ̃ (() () ) ̃ 0 (4.19)

Here ()−() is aggregate gross investment, () For a given level of ()the resulting amount of new capital goods per time unit (̇()+()), mea-sured in efficiency units, is independent of when this investment occurs. It isthereby not affected by technological progress. Similarly, the interpretationof ̃ 0 in (4.19) is that the higher technology level obtained as timeproceeds results in higher productivity of all capital and labor. Thus also

11As to there is of course a trivial exception, namely the case where the productionfunction is Cobb-Douglas and therefore is a given parameter.

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4.5. What if technological change is embodied? 71

firms that have only old capital equipment benefit from recent advances intechnical knowledge. No new investment is needed to take advantage of therecent technological and organizational developments.12

In contrast, we say that technological change is embodied, if taking ad-vantage of new technical knowledge requires construction of new investmentgoods. The newest technology is incorporated in the design of newly pro-duced equipment; and this equipment will not participate in subsequenttechnological progress. Whatever the source of new technical knowledge,investment becomes an important bearer of the productivity increases whichthis new knowledge makes possible. Without new investment, the potentialproductivity increases remain potential instead of being realized.As also noted in Chapter 2, we may represent embodied technological

progress (also called investment-specific technological change) by writing cap-ital accumulation in the following way,

̇() = ()()− () (4.20)

where () is gross investment at time and () measures the “quality”(productivity) of newly produced investment goods. The increasing level oftechnology implies increasing () so that a given level of investment givesrise to a greater and greater additions to the capital stock, measuredin efficiency units. As in our aggregate framework, capital goods can beproduced at the same minimum cost as one consumption good, we have · =1 where is the equilibrium price of capital goods in terms of consumptiongoods. So embodied technological progress is likely to result in a steadydecline in the relative price of capital equipment, a prediction confirmed bythe data (see, e.g., Greenwood et al., 1997).This raises the question how the propositions 1, 2, and 3 fare in the case

of embodied technological progress. The answer is that a generalized versionof Proposition 1 goes through. Essentially, we only need to replace (4.7) by(13.13) and interpret in Proposition 1 as the value of the capital stock,i.e., we have to replace by ̃ = But the concept of Harrod-neutrality no longer fits the situation with-

out further elaboration. Hence to obtain analogies to Proposition 2 andProposition 3 is a more complicated matter. Suffice it to say that with em-bodied technological progress, the class of production functions that are con-sistent with balanced growth is smaller than with disembodied technologicalprogress.

12In the standard versions of the Solow model and the Ramsey model it is assumed thatall technological progress has this form - for no other reason than that this is by far thesimplest case to analyze.

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72CHAPTER 4. SKILL-BIASED TECHNICAL CHANGE.

BALANCED GROWTH THEOREMS

4.6 Concluding remarks

In the Solow model as well as in many other models with disembodied techno-logical progress, a steady state and a balanced growth path imply each other.Indeed, they are in that model, as well as many others, two sides of the sameprocess. There exist exceptions, however, that is, cases where steady stateand a balanced growth are not equivalent (some open economy models andsome models with embodied technical change). So the two concepts shouldbe held apart.13

Note that the definition of balanced growth refers to aggregate variables.At the same time as there is balanced growth at the aggregate level, structuralchange may occur. That is, a changing sectorial composition of the economyis under certain conditions compatible with balanced growth (in a generalizedsense) at the aggregate level, cf. the “Kuznets facts” (see Kongsamut et al.,2001, and Acemoglu, 2009, Chapter 20).In view of the key importance of Harrod-neutrality, a natural question is:

has growth theory uncovered any endogenous tendency for technical progressto converge to Harrod-neutrality? Fortunately, in his Chapter 15 Acemogluoutlines a theory about a mechanism entailing such a tendency, the theory of“directed technical change”. Jones (2005) suggests an alternative mechanism.

4.7 References

Acemoglu, D., 2009, Introduction to Modern Economic Growth, PrincetonUniversity Press: Oxford.

Barro, R., and X. Sala-i-Martin, 2004, Economic Growth, second edition,MIT Press: Cambridge (Mass.)

Duffy. J., C. Papageorgiou, and F. Perez-Sebastian, 2004, Capital-SkillComplementarity? Evidence from a Panel of Countries, The Review ofEconomics and Statistics, vol. 86(1), 327-344.

Gordon, R. J., 1990. The Measurement of Durable goods Prices. ChicagoUniversity Press: Chicago.

13Here we deviate from Acemoglu, p. 65, where he says that he will use the two terms“interchangingly”. We also deviate from Barro and Sala-i-Martin (2004, pp. 33-34) whodefine a steady state as synonymous with a balanced growth path as the latter was definedabove.

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4.7. References 73

Greenwood, J., Z. Hercowitz, and P. Krusell, 1997. Long-Run Implicationsof Investment-Specific Technological Change. American Economic Re-view 87 (3), 342-362.

Groth, C., K.-J. Koch, and Thomas Steger, 2010, When growth is less thanexponential, Economic Theory 44, 213-242.

Groth, C., and R. Wendner, 2014. Embodied Learning by Investing andSpeed of Convergence, J. of Macroeconomics (forthcoming).

Jones, C. I., 2005, The shape of production functions and the direction oftechnical change. Quarterly Journal of Economics, no. 2, 517-549.

Jones, C. I., and D. Scrimgeour, 2008, The steady-state growth theorem:Understanding Uzawa (1961), Review of Economics and Statistics 90(1), 180-182.

Jones, C. I., and P. M. Romer, 2010, The new Kaldor facts: Ideas, insti-tutions, population, and human capital, American Economic Journal:Macroeconomics, vol. 2 (1), 224-245. Cursory.

Kongsamut, P., S. Rebelo, and D. Xie, 2001, Beyond balanced growth.Review of Economic Studies 48, 869-882.

Perez-Sebastian, F., 2008, “Testing capital-skill complementarity across sec-tors in a panel of Spanish regions”, WP 2008.

Schlicht, E., 2006, A variant of Uzawa’s theorem, Economics Bulletin 6,1-5.

Stokey, N.L., 1996, Free trade, factor returns, and factor accumulation, J.Econ. Growth, vol. 1 (4), 421-447.

Uzawa, H., 1961, Neutral inventions and the stability of growth equilibrium,Review of Economic Studies 28, No. 2, 117-124.

Wan, H. Y. Jr., 1971, Economic Growth, Harcourt Brace: New York.

c° Groth, Lecture notes in Economic Growth, (mimeo) 2015.

74CHAPTER 4. SKILL-BIASED TECHNICAL CHANGE.

BALANCED GROWTH THEOREMS

c° Groth, Lecture notes in Economic Growth, (mimeo) 2015.

Chapter 5

Growth accounting and theconcept of TFP: Some warnings

5.1 Introduction

This chapter discusses the concepts of Total Factor Productivity, TFP, andTFP growth, and ends up with three warnings regarding uncritical use ofthem.First, however, we should provide a precise definition of the TFP level

which is in fact a tricky concept. Unfortunately, Acemoglu (p. 78) doesnot make a clear distinction between TFP level and TFP growth. Moreover,Acemoglu’s point of departure (p. 77) assumes a priori that the way the pro-duction function is time-dependent can be represented by a one-dimensionalindex, () The TFP concept and the applicability of growth accountingare, however, not limited to this case.For convenience, we treat time as continuous (although the timing of the

variables is indicated merely by a subscript).1

5.2 TFP level and TFP growth

Let denote aggregate output (value added in fixed prices) at time in asector or the economy as a whole. Suppose is determined by the function

= ( ) (5.1)

1I thank Niklas Brønager for useful discussions related to this chapter.

75

76CHAPTER 5. GROWTH ACCOUNTING AND THE CONCEPT

OF TFP: SOME WARNINGS

where is an aggregate input of physical capital and an index of quality-adjusted labor input.2 The “quality-adjustment” of the input of labor (man-hours per year) aims at taking educational level and work experience intoaccount. In fact, both output and the two inputs are aggregates of het-erogeneous elements. The involved conceptual and measurement difficultiesare huge and there are different opinions in the growth accounting literatureabout how to best deal with them. Here we ignore these problems. Thethird argument in (5.1) is time, indicating that the production function (· · ) is time-dependent. Thus “shifts in the production function”, dueto changes in efficiency and technology (“technical change” for short), canbe taken into account. We treat time as continuous and assume that isa neoclassical production function. When the partial derivative of w.r.t.the third argument is positive, i.e., 0 technical change amountsto technical progress. We consider the economy from a purely supply-sideperspective.3

We shall here concentrate on the fundamentals of TFP and TFP growth.These can in principle be described without taking the heterogeneity andchanging quality of the labor input into account. Hence we shall from nowon ignore this aspect and simplifying assume that labor is homogeneous andlabor quality is constant. So (5.1) is reduced to the simpler case,

= ( ) (5.2)

where is the number of man-hours per year. As to measurement of, some adaptation of the perpetual inventory method4 is typically used,with some correction for under-estimated quality improvements of invest-ment goods in national income accounting. The output measure is (or atleast should be) corrected correspondingly, also for under-estimated qualityimprovements of consumption goods.

2Natural resources (land, oil wells, coal in the ground, etc.) constitute a third primaryproduction factor. The role of this factor is in growth accounting often subsumed under.

3Sometimes in growth accounting the left-hand side variable, in (5.2) is the grossproduct rather than value added. Then non-durable intermediate inputs should be takeninto account as a third production factor and enter as an additional argument of ̃ in(5.2). Since non-market production is difficult to measure, the government sector is usuallyexcluded from in (5.2). Total Factor Productivity is by some authors called MultifactorProductivity and abbreviated MFP.

4Cf. Section 2.2 in Chapter 2.

c° Groth, Lecture notes in Economic Growth, (mimeo) 2015.

5.2. TFP level and TFP growth 77

5.2.1 TFP growth

The notion of Total Factor Productivity at time TFP is intended toindicate a level of productivity. Nevertheless there is a tendency in theliterature to evade a direct definition of this level and instead go straightaway to a decomposition of output growth. Let us start the same way herebut not forget to come back to the issue about what can be meant by thelevel of TFP.The growth rate of a variable at time will be denoted . We take

the total derivative w.r.t. in (5.2) to get

̇ = ( )̇ + ( )̇ + ( ) · 1Dividing through by gives

≡ ̇=1

h( )̇ + ( )̇ + ( ) · 1

i=

( )

+

( )

+

( )

≡ + + ( )

(5.3)

where and are shorthands for ( ) ≡ () () and ( )≡ ()

() respectively, that is, the partial output elasticities w.r.t. the

two production factors, evaluated at the factor combination ( ) at time Finally, ( ) ≡ , that is, the partial derivative w.r.t. thethird argument of the function , evaluated at the point ( )The equation (5.3) is the basic growth-accounting relation, showing how

the output growth rate can be decomposed into the “contribution” fromgrowth in each of the inputs and a residual. The TFP growth rate is definedas the residual

TFP, ≡ − ( + ) = ( )

(5.4)

So the TFP growth rate is what is left when from the output growth rate issubtracted the “contribution” from growth in the factor inputs weighted bythe output elasticities w.r.t. these inputs. This is sometimes interpreted asreflecting that part of the output growth rate which is explained by technicalprogress. One should be careful, however, not to identify a descriptive ac-counting relationship with deeper causality. Without a complete model, atmost one can say that the TFP growth rate measures that fraction of outputgrowth that is not directly attributable to growth in the capital and laborinputs. So:

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78CHAPTER 5. GROWTH ACCOUNTING AND THE CONCEPT

OF TFP: SOME WARNINGS

The TFP growth rate can be interpreted as reflecting the “directcontribution” to current output growth from current technicalchange (in a broad sense including learning by doing and organi-zational improvement).

Let us consider how the actual measurement of TFP, can be carried out.The output elasticities w.r.t. capital and labor, and will, underperfect competition and absence of externalities and of increasing returnsto scale, equal the income shares of capital and labor, respectively. Timeseries for these income shares and for , and hence also for and , can be obtained (directly or with some adaptation) from nationalincome accounts. This allows straightforward measurement of the residual,TFP,

5

The decomposition in (5.4) was introduced already by Solow (1957). Sincethe TFP growth rate appears as a residual, it is sometimes called the Solowresidual. As a residual it may reflect the contribution of many things, somewanted (current technical innovation in a broad sense including organiza-tional improvement), others unwanted (such as varying capacity utilization,omitted inputs, measurement errors, and aggregation bias).

5.2.2 The TFP level

Now let us consider the level of TFP, that “something” for which we havecalculated its growth rate without yet having defined what it really is. Butknowing the growth rate of TFP for all in a certain time interval, we in facthave a differential equation in the TFP level of the form () = ()()namely:

(TFP) = TFP, ·TFPThe solution of this simple linear differential equation is6

TFP = TFP0 0 TFP, (5.5)

For a given initial value TFP0 0 (which may be normalized to 1 if de-sired), the time path of TFP is determined by the right-hand side of (5.5).Consequently:

The TFP level at time can interpreted as reflecting the cumula-tive “direct contribution” to output since time 0 from cumulativetechnical change since time 0.

5Of course, data are in discrete time. So to make actual calculations we have to translate(5.4) into discrete time. The weights and can then be estimated by two-yearsmoving averages of the factor income shares as shown in Acemoglu (2009, p. 79).

6See Appendix B of Chapter 3 in these lecture notes or Appendix B to Acemoglu.

c° Groth, Lecture notes in Economic Growth, (mimeo) 2015.

5.3. The case of Hicks-neutrality* 79

Why do we say “direct contribution”? The reason is that the cumulativetechnical change since time 0 may also have an indirect effect on output,namely via affecting the output elasticities w.r.t. capital and labor, and Through this channel cumulative technical change affects the role ofinput growth for output growth. This possible indirect effect over time oftechnical change is not included in the TFP concept.To clarify the matter we will compare the TFP calculation under Hicks-

neutral technical change with that under other forms of technical change.

5.3 The case of Hicks-neutrality*

In the case of Hicks neutrality, by definition, technical change can be repre-sented by the evolution of a one-dimensional variable, and the productionfunction in (5.2) can be specified as

= ( ) = ( ) (5.6)

Here the TFP level is at any time, , identical to the level of if we normalizethe initial values of both and TFP to be the same, i.e., TFP0 = 0 0.Indeed, calculating the TFP growth rate, (5.4), on the basis of (5.6) gives

TFP, =( )

=

̇ ( )

( )=

̇≡ (5.7)

where the second equality comes from the fact that and are kept fixedwhen the partial derivative of w.r.t. is calculated. The formula (5.5) nowgives

TFP = 0 · 0 , =

The nice feature of Hicks neutrality is thus that we can write

TFP = ( )

( 0)=

( )

0 ( )= (5.8)

using the normalization 0 = 1 That is:

Under Hicks neutrality, current TFP appears as the ratio be-tween the current output level and the hypothetical output levelthat would have resulted from the current inputs of capital andlabor in case of no technical change since time 0.

So in the case of Hicks neutrality the economic meaning of the TFP levelis straightforward. The reason is that under Hicks neutrality the outputelasticities w.r.t. capital and labor, and are independent of technicalchange.

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80CHAPTER 5. GROWTH ACCOUNTING AND THE CONCEPT

OF TFP: SOME WARNINGS

5.4 The case of absence of Hicks-neutrality*

The above very intuitive interpretation of TFP is only valid under Hicks-neutral technical change. Neither under general technical change nor evenunder Harrod- or Solow-neutral technical change (unless the production func-tion is Cobb-Douglas so that both Harrod and Solow neutrality imply Hicks-neutrality), will current TFP appear as the ratio between the current outputlevel and the hypothetical output level that would have resulted from thecurrent inputs of capital and labor in case of no technical change since time0.To see this, let us return to the general time-dependent production func-

tion in (5.2). Let denote the ratio between the current output level attime and the hypothetical output level, ( 0) that would have ob-tained with the current inputs of capital and labor in case of no change inthe technology since time 0, i.e.,

≡ ( ) ( 0)

(5.9)

So can be seen as a factor of joint-productivity growth from time 0 totime evaluated at the time- input combination.If this should always indicate the level of TFP at time , the growth

rate of should equal the growth rate of TFP. Generally, it does not,however. Indeed, defining ( ) ≡ ( 0) by the rule for the timederivative of fractions7, we have

≡ ̃ ( ) ( )

− ( )( )

=1

h( )̇ + ( )̇ + ( ) · 1

i− 1( )

h( )̇ +( )̇

i= ( ) + ( ) +

( )

−(( 0) + ( 0)) (5.10)

= (( )− ( 0)) + (( )− ( 0)) + TFP,6= TFP, generally,

where TFP, is given in (5.4). Unless the partial output elasticities w.r.t.capital and labor, respectively, are unaffected by technical change, the con-clusion is that TFP will differ from our defined in (5.9). So:

7See Appendix A to Chapter 3 of these lecture notes.

c° Groth, Lecture notes in Economic Growth, (mimeo) 2015.

5.4. The case of absence of Hicks-neutrality* 81

In the absence of Hicks neutrality, current TFP does not gener-ally appear as the ratio between the current output level and thehypothetical output level that would have resulted from the cur-rent inputs of capital and labor in case of no technical changesince time 0.

A closer look at vs. TFP

As in (5.9) is the time- output arising from the time- inputs relative tothe fictional time-0 output from the same inputs, we consider along withTFP as two alternative joint-productivity indices. From (5.10) we see that

TFP, = −(( )− ( 0)) −(( )−( 0))So the growth rate of TFP equals the growth rate of the joint-productivityindex corrected for the cumulative impact of technical change since time 0on the direct contribution to time- output growth from time- input growth.This impact comes about when the output elasticities w.r.t. capital and la-bor, respectively, are affected by technical change, that is, when ( )6= ( 0) and/or ( ) 6= ( 0)Under Hicks-neutral technical change there will be no correction because

the output elasticities are independent of technical change. In this case TFPcoincides with the index In the absence of Hicks-neutrality the two indicesdiffer. This is why we in Section 2.2 characterized the TFP level as thecumulative “direct contribution” to output since time 0 from cumulativetechnical change, thus excluding the possible indirect contribution comingabout via the potential effect of technical change on the output elasticitiesw.r.t. capital and labor and thereby on the contribution to output from inputgrowth.Given that the joint-productivity index is the more intuitive joint-

productivity measure, why is TFP the more popular measure? There are atleast two reasons for this. First, it can be shown that the TFP measure hasmore convenient balanced growth properties. Second, is more difficult tomeasure. To see this we substitute (5.3) into (5.10) to get

= − (( 0) + ( 0)) (5.11)

The relevant output elasticities, ( 0)≡ (0) (0) and ( 0)≡ (0)

(0) are hypothetical constructs, referring to the technology as it

was at time 0, but with the factor combination observed at time , not at time0. The nice thing about the Solow residual is that under the assumptionsof perfect competition and absence of externalities, it allows measurement

c° Groth, Lecture notes in Economic Growth, (mimeo) 2015.

82CHAPTER 5. GROWTH ACCOUNTING AND THE CONCEPT

OF TFP: SOME WARNINGS

by using data on prices and quantities alone, that is, without knowledgeof the production function. To evaluate , however, we need estimatesof the hypothetical output elasticities, ( 0) and ( 0) Thisrequires knowledge about how the output elasticities depend on the factorcombination and time, respectively, that is, knowledge about the productionfunction.Now to the warnings concerning application of the TFP measure.

5.5 Three warnings

Balanced growth at the aggregate level, hence Harrod neutrality, seems tocharacterize the growth experience of the UK and US over at least a century(Kongsamut et al., 2001; Attfield and Temple, 2010). At the same timethe aggregate elasticity of factor substitution is generally estimated to besignificantly less than one (see, e.g., Antras, 2004). This amounts to rejectionof the Cobb-Douglas specification of the aggregate production function andso, at the aggregate level, Harrod neutrality rules out Hicks neutrality.

Warning 1 Since Hicks-neutrality is empirically doubtful at the aggre-gate level, TFP can often not be identified with the simple intuitive joint-productivity measure defined in (5.9) above.

Warning 2 When Harrod neutrality obtains, relative TFP growth ratesacross sectors or countries can be quite deceptive.Suppose there are countries and that country has the aggregate pro-

duction function

= ()( ) = 1 2

where () is a neoclassical production function with CRS and is the levelof labor-augmenting technology which, for simplicity, we assume shared byall the countries (these are open and “close” to each other). So technicalprogress is Harrod-neutral. Let the growth rate of be a constant 0Many models imply that ̃ ≡ () tends to a constant, ̃∗ , in the longrun, which we assume is also the case here. Then, for →∞ ≡ ≡ ̃ where ̃ → ̃∗ and ≡ ≡ ̃ where ̃ → ̃∗ = ()(̃∗ );here () is the production function on intensive form. So in the long run and tend to = .

c° Groth, Lecture notes in Economic Growth, (mimeo) 2015.

5.5. Three warnings 83

Formula (5.4) then gives the TFP growth rate of country in the longrun as

TFP = − (∗ + (1− ∗ )) = − − ∗ ( − )= − ∗ = (1− ∗ ) (5.12)

where ∗ is the output elasticity w.r.t. capital, ()0(̃)̃ ()(̃) evaluated

at ̃ = ̃∗ Under labor-augmenting technical progress, the TFP growth ratethus varies negatively with the output elasticity w.r.t. capital (the capitalincome share under perfect competition). Owing to differences in productand industry composition, the countries have different ∗ ’s. In view of (5.12),for two different countries, and we get

→⎧⎨⎩∞ if ∗ ∗ 1 if ∗ =

∗

0 if ∗ ∗

(5.13)

for → ∞8 Thus, in spite of long-run growth in the essential variable, being the same across the countries, their TFP growth rates are verydifferent. Countries with low ∗ ’s appear to be technologically very dynamicand countries with high ∗ ’s appear to be lagging behind. It is all due to thedifference in across countries; a higher just means that a larger fractionof = = becomes “explained” by in the growth accounting (5.12),leaving a smaller residual. And the level of has nothing to do with technicalprogress.We conclude that comparison of TFP levels across countries or time may

misrepresent the intuitive meaning of productivity and technical progresswhen output elasticities w.r.t. capital differ and technical progress is Harrod-neutral (even if technical progress were at the same time Hicks-neutral as isthe case with a Cobb-Douglas specification). It may be more reasonable tojust compare levels of across countries and time.

Warning 3 Growth accounting is - as the name says - just about account-ing and measurement. So do not confuse growth accounting with causalityin growth analysis. To talk about causality we need a theoretical model sup-ported by the data. On the basis of such a model we can say that this or thatset of exogenous factors through the propagation mechanisms of the modelcause this or that phenomenon, including economic growth. In contrast, con-sidering the growth accounting identity (5.3) in itself, none of the terms have

8If is Cobb-Douglas with output elasticity w.r.t. capital equal to , the resultin (5.12) can be derived more directly by first defining =

1− , then writing the

production function in the Hicks-neutral form (5.6), and finally use (5.7).

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84CHAPTER 5. GROWTH ACCOUNTING AND THE CONCEPT

OF TFP: SOME WARNINGS

priority over the others w.r.t. a causal role. And there are important omittedvariables. There are simple illustrations in Exercises III.1 and III.2.In a complete model with exogenous technical progress, part of will

be induced by this technical progress. If technical progress is endogenousthrough learning by investing, as in Arrow (1962), there is mutual causa-tion between and technical progress. Yet another kind of model mightexplain both technical progress and capital accumulation through R&D, cf.the survey by Barro (1999).

5.6 References

Antràs, P., 2004, Is the U.S. aggregate production function Cobb-Douglas?New estimates of the elasticity of substitution, Contributions to Macro-economics, vol. 4, no. 1, 1-34.

Attfield, C., and J.R.W. temple, 2010, Balanced growth and the great ratios:New evidence for the US and UK, J. of Macroeconomics, vol. 32, 937-956.

Barro, R.J., 1999, Notes on growth accounting, J. of Economic Growth, vol.4 (2), 119-137.

Bernard, A. B., and C. I. Jones, 1996a, Technology and Convergence, Eco-nomic Journal, vol. 106, 1037-1044.

Bernard, A. B., and C. I. Jones, 1996b, Comparing Apples to Oranges:productivity convergence and measurement across industries and coun-tries, American Economic Review, vol. 86, no. 5, 1216-1238.

Greenwood, J., and P. Krusell, 2006, Growth accounting with investment-specific technological progress: A discussion of two approaches, J. ofMonetary Economics.

Hercowitz, Z., 1998, The ‘embodiment’ controversy: A review essay, J. ofMonetary Economics, vol. 41, 217-224.

Hulten, C.R., 2001, Total factor productivity. A short biography. In: Hul-ten, C.R., E.R. Dean, and M. Harper (eds.), New Developments inProductivity Analysis, Chicago: University of Chicago Press, 2001, 1-47.

Kongsamut, P., S. Rebelo, and D. Xie, 2001, Beyond Balanced Growth,Review of Economic Studies, vol. 68, 869-882.

c° Groth, Lecture notes in Economic Growth, (mimeo) 2015.

5.6. References 85

Sakellaris, P., and D.J. Wilson, 2004, Quantifying embodied technologicalprogress, Review of Economic Dynamics, vol. 7, 1-26.

Solow, R.M., 1957, Technical change and the aggregate production function,Review of Economics and Statistics, vol. 39, 312-20.

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86CHAPTER 5. GROWTH ACCOUNTING AND THE CONCEPT

OF TFP: SOME WARNINGS

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